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# I.S.I. 2015 M.Math Entrance Subjective Problems

If you know any other problem, please put in the comment section.

1. Let $$(a_k)$$ be a sequence in (0, 1) . Prove that $$a_k to 0$$ iff $$\displaystyle{ \frac{1}{n} \sum _{k=1}^n to 0 }$$
2. Let $$f: setR to setR$$ be a continuously differentiable function such that $$\infty_{x \in setR} f'(x) > 0$$ . Prove that there exist some $$a \in set R$$ such that f(a) = 0.
3. Let X be a metric space such that $$A \subset X$$, A is an uncountable subset of X . Prove that X is not separable. Given {d(x, y) : $$x \in A , y \in A$$ } > 0
4. Let the differential equation be y” + P(x) y’ + Q(x) y = R(x)  where P, Q, R are continuous functions on [a, b] .
May 10, 2015